Carleson measures in Hardy and weighted Bergman spaces of polydiscs

Jafari, F.

doi:10.1090/S0002-9939-1991-1039533-0

Carleson measures in Hardy and weighted Bergman spaces of polydiscs
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Proc. Amer. Math. Soc. 112 (1991), 771-781 Request permission

Abstract:

The importance of theorems on Carleson measures has been well recognized [3]. In [1] Chang has given a characterization of the bounded measures on ${L^p}({T^n})$ using what one may characterize as the bounded identity operators from Hardy spaces of polydiscs in ${L^p}$ spaces. In [4] Hastings gives a similar result for (unweighted) Bergman spaces of polydiscs. In this paper we characterize the bounded identity operators from weighted Bergman spaces of polydiscs into ${L^p}$ spaces, and classify those operators which are compact on the Hardy and weighted Bergman spaces in terms of Carleson-type conditions. We give two immediate applications of these results here, and a much broader class of applications elsewhere [5].

References

Sun-Yung A. Chang, Carleson measure on the bi-disc, Ann. of Math. (2) 109 (1979), no. 3, 613–620. MR 534766, DOI 10.2307/1971229
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
William W. Hastings, A Carleson measure theorem for Bergman spaces, Proc. Amer. Math. Soc. 52 (1975), 237–241. MR 374886, DOI 10.1090/S0002-9939-1975-0374886-9
F. Jafari, On bounded and compact composition operators in polydiscs, Canad. J. Math. 42 (1990), no. 5, 869–889. MR 1081000, DOI 10.4153/CJM-1990-045-0

Composition operators in polydiscs

Barbara D. MacCluer, Compact composition operators on $H^p(B_N)$, Michigan Math. J. 32 (1985), no. 2, 237–248. MR 783578, DOI 10.1307/mmj/1029003191
Barbara D. MacCluer and Joel H. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38 (1986), no. 4, 878–906. MR 854144, DOI 10.4153/CJM-1986-043-4
Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
David A. Stegenga, Multipliers of the Dirichlet space, Illinois J. Math. 24 (1980), no. 1, 113–139. MR 550655